Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #7391 |
116 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 2 generators on 143 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(80,82,87,99)(81,88,94,117)(83,101,104,138)(84,105,85,110)(86,112,107,98)(89,119,124,142)(90,114,109,126)(91,113,120,143)(92,116,93,115)(95,125,130,121)(96,131,136,139)(97,111,132,108)(100,129,137,128)(102,135,106,140)(103,134,118,133)(122,141,123,127);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81,84,90,86,92,108,120,87,94,85,109,107,93,111,91)(82,95,97,119,98,128,105,127,99,130,132,142,112,129,110,141)(83,100,102,124,103,125,139,123,104,137,106,89,118,121,131,122)(88,101,113,96,115,133,126,135,117,138,143,136,116,134,114,140)Polycyclic presentation:
2^8,79|a^2=d,b^2=h,c^2=eh,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^78,c^b=ceg,d^b=dg,e^b=egh,f^b=fh,g^b=gh,i^b=i^78,d^c=dh1750 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57043);